Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 217280, 16 pages
doi:10.1155/2012/217280
Research Article
A Note on the Growth of Meromorphic Functions
with a Radially Distributed Value
Jun-Fan Chen
Department of Mathematics, Fujian Normal University, Fujian Province, Fuzhou 350007, China
Correspondence should be addressed to Jun-Fan Chen, junfanchen@163.com
Received 8 July 2012; Accepted 17 October 2012
Academic Editor: Ferhan M. Atici
Copyright q 2012 Jun-Fan Chen. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This note is to investigate the growth of transcendental meromorphic functions with radially
distributed values. We generalize a more recent result of Chen et al. 2011. The paper is closely
related to some previous results due to Fang and Zalcman 2008, and Xu et al. 2009.
1. Introduction and Results
be a meromorphic function, where C is the whole complex plane and C
Let f: C → C
C ∪ {∞}. We shall use the basic results and notations of Nevanlinna’s value distribution
theory of meromorphic functions see 1–3, such as T r, f, Nr, f, and mr, f. Meantime,
the Nevanlinna’s deficiency δa, f of fz with respect to a ∈ C is defined by
m r, 1/ f − a
N r, 1/ f − a
1 − lim sup
δ a, f lim inf
r →∞
T r, f
T r, f
r →∞
1.1
and δ∞, f is obtained by the above formula with mr, f in place of mr, 1/f − a, and
Nr, f in place of Nr, 1/f − a, respectively. The lower order μ and order λ are defined in
turn as follows:
log T r, f
μ : μ f lim inf
,
r →∞
log r
log T r, f
λ : λ f lim sup
.
log r
r →∞
1.2
2
Abstract and Applied Analysis
we denote by nr, X, f a the number of
For an unbounded subset X of C and a ∈ C,
the zeros of f − a counting multiplicities in X ∩ {z : |z| ≤ r}.
In 1992, Yang 4, cf. 5 posed the following interesting conjecture.
Yang’s Conjecture
Let P be a property or a set of properties such that any entire or meromorphic in C
function satisfying P must be a constant. Suppose that f is an entire or meromorphic in
C function of finite lower order μ, and that Lj : arg z θj j 1, 2, . . . , q; 0 ≤ θ1 < θ2 < · · · <
θq < 2π; θq1 θ1 2π are a finite number of rays issuing from the origin. If f satisfies P in
q
C \ j1 Lj , then the order λ of f has the following estimation:
π
λ ≤ max
, θq1 θ1 2π .
1≤j≤q θj1 − θj
1.3
Yang’s conjecture implies that if meromorphic functions satisfy certain properties in
the vicinities of a finite number of rays, then the growth of meromorphic functions will be
restricted.
A well-known result of Clunie 6, cf. 7 is that an entire function f which satisfies
1 in C must be constant. In 5, L. Yang and C.-C. Yang chose the property P as
fzf z /
1 and verified the above conjecture. On the other hand, Fang and Zalcman 8
fzf z /
considered the value distribution of f af n in C, where a is a nonzero finite complex
number and n ≥ 2 is a positive integer. Actually, they 8 gave an affirmative answer to
a question suggested by Ye 9. Later on, Xu et al. 10 further generalized f af n to
f af k n in C for a positive integer k with n ≥ k 1 and investigated its value distribution.
More recently, Chen et al. 11 chose another property P as f af n to continue to
study Yang’s Conjecture and proved the following results.
Theorem A. Let fz be a transcendental meromorphic function with δ∞, f > 0 in C and let
Lj : arg z θj j 1, 2, . . . , q be a finite number of rays issued from the origin such that
−π ≤ θ1 < θ2 < · · · < θq < π,
with ω max{π/θj1 − θj : 1 ≤ j ≤ q}. Set Y C \ θq1 θ1 2π
q
j1
1.4
Lj . If fz satisfies
n
log n r, Y, f a f b
lim sup
≤ρ
log r
r →∞
1.5
with a positive number ρ, finite complex numbers a /
0 and b, for any positive integer n ≥ 2, then the
order λ of fz has the estimation λ ≤ max{ω, ρ}.
Theorem B. Let fz be a transcendental meromorphic function of finite lower order μ with
δ∞, f > 0 in C. For q pairs of real numbers {αj , βj } such that
−π ≤ α1 < β1 ≤ α2 < β2 ≤ · · · ≤ αq < βq ≤ π
1.6
Abstract and Applied Analysis
3
with ω max{π/βj − αj : 1 ≤ j ≤ q}, suppose that
n
log n r, Y, f a f b
≤ρ
lim sup
log r
r →∞
with a positive number ρ, finite complex numbers a /
0 and b, and Y any positive integer n ≥ 2, and that
q
αj1 − βj
j1
δ ∞, f 4
,
< arcsin
σ
2
1.7
q
j1 {z
: αj ≤ arg z ≤ βj }, for
αq1 α1 2π
1.8
with σ max{ω, ρ, μ}. Then the order λ of fz has the estimation λ ≤ max{ω, ρ}.
Now there arises a natural question.
Question 1. What can be said if f in Theorems A and B is replaced by the kth derivative f k ?
In this paper, we will prove the following results which generalize Theorems A and B.
Theorem 1.1. Let fz be a transcendental meromorphic function with δ∞, f k > 0 for a positive
integer k in C and let Lj : arg z θj j 1, 2, . . . , q be a finite number of rays issued from the origin
such that
−π ≤ θ1 < θ2 < · · · < θq < π,
with ω max{π/θj1 − θj : 1 ≤ j ≤ q}. Set Y C \ lim sup
r →∞
θq1 θ1 2π
q
j1
1.9
Lj . If fz satisfies
n
log n r, Y, f a f k b
log r
≤ρ
1.10
with a positive number ρ, finite complex numbers a / 0 and b, for any positive integer n ≥ k 1, then
the order λ of fz has the estimation λ ≤ max{ω, ρ}.
Remark 1.2. Let k 1. Then by Theorem 1.1 we get Theorem A.
Corollary 1.3. Let fz be a transcendental entire function, let the notations θj (j 1, 2, . . . , q 1),
ω, and Y be defined as in Theorem 1.1, and suppose that the function fz fulfills the same condition
1.10 as in Theorem 1.1. Then the order λ of fz has the estimation λ ≤ max{ω, ρ}.
Theorem 1.4. Let fz be a transcendental meromorphic function of finite lower order μ with
δ∞, f k > 0 for a positive integer k in C. For q pairs of real numbers {αj , βj } such that
−π ≤ α1 < β1 ≤ α2 < β2 ≤ · · · ≤ αq < βq ≤ π
1.11
4
Abstract and Applied Analysis
with ω max{π/βj − αj : 1 ≤ j ≤ q}, suppose that
lim sup
n
log n r, Y, f a f k b
r →∞
log r
with a positive number ρ, finite complex numbers a /
0 and b, and Y any positive integer n ≥ k 1, and that
q
αj1 − βj
j1
4
< arcsin
σ
δ ∞, f k
,
2
1.12
≤ρ
q
j1 {z
: αj ≤ arg z ≤ βj }, for
αq1 α1 2π
1.13
with σ max{ω, ρ, μ}. Then the order λ of fz has the estimation λ ≤ max{ω, ρ}.
Remark 1.5. Let k 1. Then by Theorem 1.4 we get Theorem B.
Corollary 1.6. Let fz be a transcendental entire function of finite lower order μ, let the notations
αj , βj (j 1, 2, . . . , q), ω, σ, and Y be defined as in Theorem 1.4, and suppose that the function fz
fulfills the same conditions 1.12 and 1.13 as in Theorem 1.4. Then the order λ of fz has the
estimation λ ≤ max{ω, ρ}.
In order to prove our results, we require the Nevanlinna theory of meromorphic
functions in an angular domain. For the sake of convenience, we recall some notations and
definitions. Let f be a meromorphic function on the angular domain Ωα, β {z : α ≤ arg z ≤
β}, where 0 < β − α ≤ 2π. Nevanlinna et al. 12, 13 introduced the following notations:
r
dt
1
tω iα ,
log f te log f teiβ −
ω
2ω
t
t
r
1
2ω β
iθ Bα,β r, f re
log
sin ωθ − αdθ,
f
πr ω α
1
|bm |ω
Cα,β r, f 2
−
sin ωθm − α,
r 2ω
|bm |ω
1<|bm |<r
ω
Aα,β r, f π
1.14
where ω π/β − α and bm |bm |eiθm are the poles of f in Ωα, β appearing according to
their multiplicities. The function Cα,β r, f is called the angular counting function counting
multiplicities of the poles of f in Ωα, β, and Cα,β r, f is called the angular reduced
counting function ignoring multiplicities of the poles of f in Ωα, β. Further, Nevanlinna’s
angular characteristic function Sα,β r, f is defined as follows:
Sα,β r, f Aα,β r, f Bα,β r, f Cα,β r, f .
1.15
Throughout the paper, we denote by Rr, ∗ a quantity satisfying
Rr, ∗ O logrT r, ∗ ,
∀r ∈
/ E,
1.16
Abstract and Applied Analysis
5
where E denotes a set of positive real numbers with finite linear measure. It is not necessarily
the same for every occurrence in the context.
2. Some Lemmas
In this section we present some lemmas which will be needed in the sequel.
Lemma 2.1 see 5, 12–14. Let f be meromorphic in C. Then in Ωα, β for an arbitrary finite
complex number a, we have
1
Sα,β r, f O1,
Sα,β r,
2.1
f −a
and for each positive integer k, we have
f k
r,
f
Aα,β
f k
r,
f
Bα,β
R r, f .
2.2
Lemma 2.2 see 10. Let n ≥ 2 and k be positive integers, let a be a nonzero finite complex number,
and let pz be a polynomial. Then the solution of the differential equation aωk zn ωz pz
must be polynomial.
Lemma 2.3 see 1, Theorem 3.1; 1, page 33, 2.1. Let f be meromorphic in C, let
a1 , a2 , . . . , aq , where q ≥ 2, be distinct complex numbers, δ > 0, and suppose that |aμ − aν | ≥ δ
for 1 ≤ μ < ν ≤ q. Then, for each positive integer k, we have
T r, f k ≤ T r, f kN r, f O log rT r, f ,
q
1
m r,
f
−
aν
ν1
≥
q
m r,
ν1
1
f − aν
− qlog
3q
− log 2,
δ
2.3
∀r ∈
/ E.
Next we slightly modify the proof of Lemma 2.4 in [10] to give the following key lemma, which is an
important generalization of Lemma 3 in [11].
Lemma 2.4. Let f be transcendental meromorphic in C, let a / 0 and b be finite complex numbers,
and let n and k be positive integers with n ≥ k 1. Then in Ωα, β,
Bα,β r, f
k
≤ k 1Cα,β r,
1
n
f a f k − b
R r, f .
2.4
Proof. Put
n
g f a f k − b,
φ
g k
.
g
2.5
2.6
6
Abstract and Applied Analysis
Then φ /
≡ 0, for otherwise g would be a polynomial of degree at most k − 1. This and
2.5 together with Lemma 2.2 imply that f must be a polynomial, a contradiction. By the
Nevanlinna’s basic reasoning, Lemmas 2.1, and 2.3, 2.5, and 2.6, we have
Aα,β Bα,β
r, φ R r, g R r, f .
2.7
Now a simple calculation for 2.5 shows that
n k
g k f k a f k
f k 1 Q f k ,
2.8
where Qf k is a homogeneous differential polynomial in f k of degree n − 1 and of the
form
n−k−1
Q f k a f k
⎞
⎛ k
k−2
k−1
n! f k1
kk − 1n! k k1
f
f
×⎝
f k2 · · · n f k
f 2k ⎠.
2!n − k 1!
n − k!
2.9
Then by 2.5, 2.6, and 2.8, we get
n
f k 1 Q f k
φ f a f k − b .
2.10
From 2.5–2.10, and Lemma 2.1, it thus follows that
Cα,β r,
1
f k
≤ Cα,β
Cα,β r,
1
r,
φ
1
Q f k 1
Cα,β r,
1
n
f a f k − b
1
R r, f
≤ Cα,β r, φ Cα,β r,
k n
f a f
−b
1
≤ kCα,β r, f k 1Cα,β r,
R r, f
k n
f a f
−b
1
k
k
k 1Cα,β r,
≤
Cα,β r, f
R r, f .
k n
k1
f a f
−b
2.11
Abstract and Applied Analysis
7
On the other hand, by the Nevanlinna’s basic reasoning, Lemmas 2.1 and 2.3, we
deduce that
Aα,β Bα,β
1
r, n−1
k
f
≤ Aα,β Bα,β
Aα,β Bα,β
1
r, k Q f
Aα,β Bα,β
r,
≤ Aα,β Bα,β
≤ Aα,β Bα,β
r,
1
Q f k 1
Q f k
r, n−1
f k
Aα,β Bα,β
1
Q f k 1
1
1
r, k k Q f
Q f
1
R r, f
k k Q f
1
Q f
r,
Q f k
Q f k 1
Aα,β Bα,β
1
r, Q f k
R r, f
1
k
≤ Sα,β r, Q f
− Cα,β r, R r, f
k
Q f
≤ Sα,β r, Q f
≤ Sα,β r, Q f
k
k
1
Cα,β r, f − Cα,β r, Q f k
R r, f
1
1
k
R r, f .
− Cα,β r, Cα,β r, f
k1
Q f k
2.12
This, together with Lemma 2.1, yields
n − 1Sα,β r, f
k
1
1
1
k
≤
n − 1Cα,β r, k Cα,β r, k Cα,β r, f
k1
f
Q f
1
1
− Cα,β r, R r, f .
k
Q f
2.13
Next we divide into two cases.
8
Abstract and Applied Analysis
Case 1 n ≥ k 2. Suppose that z0 is a zero of f k of multiplicity m. Then we can know that
z0 is a zero of Qf k of multiplicity at least n − k − 1m km − 1 − 1 n − 1m − k − 1.
Thus, we have
1
1
1
− Cα,β r, n − 1Cα,β r, k Cα,β r, k f
Q f
1
Q f k
1
1
1
Cα,β r, − Cα,β r, 2.14
n−1 Cα,β r, k Q f
1
Q f k
f k
1
1
.
≤ k 1Cα,β r, k Cα,β r, k f
Q f
1
Substituting this into 2.13 gives
n − 1Sα,β r, f
k
1
1
k
Cα,β r, f
≤
k 1Cα,β r, k
k1
f
1
R r, f .
Cα,β r, k Q f
1
2.15
From this and 2.11 it follows that
n − 1Sα,β r, f
implying that
Sα,β r, f
k
k
1
k2 k 1
2
k
Cα,β r, f
≤
k 1 Cα,β r,
R r, f ,
k n
k1
f a f
−b
2.16
k 12
1
k2 k 1
k
Cα,β r,
≤
Cα,β r, f
R r, f .
k n
n−1
n − 1k 1
f a f
−b
2.17
Thereby, noting n ≥ k 2, we get
Sα,β r, f
so that
k
Bα,β r, f
k
R r, f
k n
f a f
−b
k 1
1
≤ Cα,β r, f k k 1Cα,β r,
R r, f ,
k n
f a f
−b
≤
k2 k 1
2
≤ Aα,β r, f
Cα,β r, f
k
k
1
k 1Cα,β r,
Bα,β r, f
k
≤ k 1Cα,β r,
1
n
f a f k − b
2.18
R r, f .
2.19
This is the desired result.
Abstract and Applied Analysis
9
Case 2 n k 1. Then, by 2.11 and 2.13, we have
kSα,β r, f
k
1
k2 1
k
Cα,β r, f
≤
kk 1Cα,β r,
R r, f ,
k n
k1
f a f
−b
2.20
k2 1
1
k
Cα,β r, f
≤
k 1Cα,β r,
R r, f
k n
kk 1
f a f
−b
1
k
k 1Cα,β r,
R r, f .
≤ Cα,β r, f
k n
f a f
−b
2.21
which leads to
Sα,β r, f
k
This yields
Bα,β r, f
k
≤ Aα,β r, f
k
Bα,β r, f
k
≤ k 1Cα,β r,
1
n
f a f k − b
R r, f ,
2.22
obtaining the desired result.
This completes the proof of Lemma 2.4.
The following auxiliary results regarding Pólya peaks and the spread relation are
necessary in the proofs of our theorems.
Lemma 2.5 see 14–16. Let f be a transcendental meromorphic function of finite lower order μ
and order λ0 < λ ≤ ∞ in C. Then, for an arbitrary positive number σ satisfying μ ≤ σ ≤ λ and any
set E of finite linear measure, there exist Pólya peaks {rn } satisfying the following:
i rn ∈
/ E, limn → ∞ rn /n ∞;
ii lim infn → ∞ log T rn , f/ log rn ≥ σ;
iii T t, f < 1 o1t/rn σ T rn , f, t ∈ rn /n, nrn .
A sequence of {rn } satisfying (i), (ii), and (iii) in Lemma 2.5 is called a Pólya peak of order σ
of f outside E. Given a positive function Λ Λr on 0, ∞ with Λ → 0 as r → ∞, we define
> ΛrT r, f ,
θ ∈ −π, π | log iθ f re − a
DΛ r, a 1
DΛ r, ∞ θ ∈ −π, π | log f reiθ > ΛrT r, f .
2.23
Lemma 2.6 see 17. Let f be a transcendental meromorphic function of finite lower order μ and
Then for an arbitrary
order λ0 < λ ≤ ∞ in C. Suppose that δ δa, f > 0 for some a ∈ C.
10
Abstract and Applied Analysis
Pólya peak {rn } of order σμ ≤ σ ≤ λ and an arbitrary positive function Λ Λr with Λ → 0 as
r → ∞, we have
⎧
⎫
⎨
4
δ⎬
arcsin
.
lim inf meas DΛ rn , a ≥ min 2π,
n→∞
⎩
σ
2⎭
2.24
Now a more precise estimation of T r, f in terms of T r, f k is introduced as follows.
Lemma 2.7 see 18. Let f be transcendental meromorphic in C. Then, for a positive integer k and
a real number τ > 1, we have
T r, f < Kτ,k T τr, f k logτr O1,
2.25
where Kτ,k is a positive number depending on only τ and k.
At last, we state the following results due to Edrei, Hayman, and Miles, respectively.
Lemma 2.8 see 19. Let f be a transcendental meromorphic function with δ δ∞, f > 0 in C.
Then, given ε > 0, we have
meas E r, f >
Tε
r, f
1
log r
!1ε ,
∀r ∈
/ F,
2.26
where
E r, f "
δ #
θ ∈ −π, π : log f reiθ > T r, f
4
and F is a set of positive real numbers with finite logarithmic measure (i.e.,
on ε only.
$
F
2.27
dt/t < ∞ depending
Lemma 2.9 see 20. Let f be a transcendental meromorphic function in C. Then for each K > 1
there exists a set MK of the lower logarithmic density at least dK 1 − 2eK−1 − 1−1 > 0, that
is,
1
log dens MK : lim inf
r → ∞ log r
MK∩1,r
dt
≥ dK,
t
2.28
such that, for every positive integer p, we have
T r, f
lim sup
p ≤ 3eK.
r → ∞, r∈MK T r, f
2.29
Abstract and Applied Analysis
11
3. Proofs of Theorems 1.1 and 1.4
In this section, we state the detailed proofs of Theorems 1.1 and 1.4 by using the method in
14. To begin with, we give the proof of Theorem 1.4. Finally the proof of Theorem 1.1 can
be derived from Theorem 1.4.
3.1. Proof of Theorem 1.4
Assume on the contrary that Theorem 1.4 does not hold. Then λf > max{ω, ρ}. Now by
1.12, we have
n
n r, Ω αj , βj , f a f k b ≤ r ρε ,
3.1
j 1, 2, . . . , q
for arbitrarily small ε > 0 and sufficiently large r ≥ r0 . Let ξm be the zeros of f af k n − b on
Ωαj , βj appearing according to their multiplicities, and set ωj π/βj −αj . By the definition
of Cα,β r, ∗, we deduce that
Cαj ,βj
r,
1
n
f a f k − b
1
2
≤2
|ξ |ωj
1<|ξm |<r m
≤2
r dn t, Ωαj , βj , f af k n b
tω j
1
n
n r, Ω αj , βj , f a f k b
r ωj
≤ 2r ρε−ωj O1 2ωj
r
r0
2ωj
r n t, Ωαj , βj , f af k n b
tωj 1
1
3.2
dt
tρε
dt
tωj 1
≤ Kj,ε r ρε−ωj O log r ,
where Kj,ε is a positive number depending on only j and ε, which is not necessarily the same
for every occurrence in the context. From Lemma 2.4, we have
Bαj ,βj r, f
k
≤ k 1Cαj ,βj
r,
1
n
f a f k − b
R r, f .
3.3
∀r ∈
/ E,
3.4
Thus, it follows by 3.2 and 3.3 that
Bαj ,βj r, f k ≤ Kj,ε r ρε−ωj O log rT r, f ,
where the exceptional set E associated with Rr, f is of at most finite linear measure.
12
Abstract and Applied Analysis
Now we discuss two cases separately.
Case 1 λf > μf. Then by the assumption σ max{ω, ρ, μ} and λf > max{ω, ρ}, we
have λf k λf > σ ≥ μf μf k . Now from 1.13, we can find a real number ε > 0
such that
q
j1
4
arcsin
αj1 − βj 2ε 2ε <
σ 2ε
δ ∞, f k
,
2
αq1 α1 2π,
λ f k > σ 2ε > μ f k .
3.5
3.6
Applying Lemma 2.5 to f k gives the existence of the Pólya peak {rn } of order σ 2ε of f k
outside the set E. Then, noting that σ 2ε > ωj ≥ 1/2 and 0 < δ∞, f k ≤ 1, by applying
Lemma 2.6 to the Pólya peak {rn }, for sufficiently large n we have
4
arcsin
meas DΛ rn , ∞ >
σ 2ε
δ ∞, f k
− ε.
2
3.7
Without loss of generality, we can assume that 3.7 holds for all the n. Set
⎛
Kn : meas⎝DΛ rn , ∞ ∩
⎞
q
%
αj ε, βj − ε ⎠.
3.8
j1
It then follows from 3.5 and 3.7 that
⎛
Kn ≥ measDΛ rn , ∞ − meas⎝−π, π \
⎞
αj ε, βj − ε ⎠
q
%
j1
⎛
measDΛ rn , ∞ − meas⎝
⎞
q
%
βj − ε, αj1 ε ⎠
3.9
j1
measDΛ rn , ∞ −
q
αj1 − βj 2ε > ε > 0.
j1
By 3.9, it is easy to see that there exists a j0 such that, for infinitely many n, we have
Kn ε
> .
meas DΛ rn , ∞ ∩ αj0 ε, βj0 − ε ≥
q
q
3.10
Abstract and Applied Analysis
13
Without loss of generality, we can assume that 3.10 holds for all the n. Set En DΛ rn , ∞ ∩
αj0 ε, βj0 − ε and Λr log r−1 . From the definition of DΛ rn , ∞, we deduce that
βj
0
−ε
αj0 ε
log f
k
rn e dθ ≥
iθ
log f k rn eiθ dθ
En
ε T rn , f k
≥
.
q log rn
3.11
On the other hand, by the definition of Bα,β r, ∗ and 3.4, it follows that
βj
0
−ε
αj0 ε
log f k rn eiθ dθ ≤
ωj
πrn 0
Bαj0 ,βj0 rn , f k
2ωj0 sin εωj0
ρε
ωj
,
≤ Kj0 ,ε rn rn 0 log rn T rn , f
3.12
where rn ∈
/ E, ωj0 π/βj0 − αj0 , and Kj0 ,ε is a positive number depending on only j0 and ε.
Combining 3.11 with 3.12 gives
qKj ,ε log rn ρε
ωj
0
,
rn rn 0 log rn T rn , f
T rn , f k ≤
ε
3.13
implying together with iii in Lemma 2.5 and Lemma 2.7 that
log T rn , f k ≤ 3 log log rn max ρ ε, ωj0 log rn log log T rn , f k O1.
3.14
Thus, from ii in Lemma 2.5 for σ 2ε, we have
σ 2ε ≤ lim sup
n→∞
log T rn , f k
≤ max ρ ε, ωj0 ≤ σ ε,
log rn
3.15
which is impossible.
Case 2 λf μf. Then by the assumption σ max{ω, ρ, μ} and λf > max{ω, ρ}, we
have σ μf λf λf k μf k . By the same argument as in Case 1 with all the
σ 2ε replaced by σ μf, we can derive
μ f σ ≤ max ρ ε, ωj0 < λ f ,
which is also impossible.
This completes the proof of Theorem 1.4.
3.16
14
Abstract and Applied Analysis
3.2. Proof of Theorem 1.1
By Theorem 1.4, it suffices to prove that the lower order μf of f is finite. As in the proof of
Theorem 1.4, we have, for each j ∈ {1, 2, . . . , q},
Bθj ,θj1 r, f k ≤ Kj,ε r ρε−ωj O log rT r, f ,
ωj π
, ∀r ∈
/ E,
θj1 − θj
3.17
where the exceptional set E associated with Rr, f is of at most finite linear measure.
For F in Lemma 2.8 and E in 3.17, log densF ∪ E 0 and hence for M2 in
Lemma 2.9 when K 2, log densM2 \ F ∪ E ≥ d2 > 0. Applying Lemma 2.8 to f k
gives the existence of a sequence {rn } of positive numbers such that rn → ∞ n → ∞,
rn ∈ M2 \ F ∪ E, and
meas E rn , f k >
ε
T rn , f
1
k
log rn
!1ε .
3.18
Set
εn 1
1
.
2q 1 T ε rn , f k log rn !1ε
3.19
Then, from 3.18 and 3.19, it follows that
⎛
meas⎝E rn , f k ∩
⎞
q
%
θj εn , θj1 − εn ⎠
j1
⎛
≥ meas E rn , f k − meas⎝
⎞
θj − εn , θj εn ⎠
q
%
3.20
j1
≥ 2q 1 εn − 2qεn εn > 0.
Hence, there exists a j ∈ {1, 2, . . . , q} such that, for infinitely many n, we have
εn
meas E rn , f k ∩ θj εn , θj1 − εn ≥ .
q
3.21
Abstract and Applied Analysis
15
Without loss of generality, we can assume that this holds for all the n. Let En Ern , f k ∩
θj εn , θj1 − εn . Thus, by the definition of Er, f and 3.21, it follows that
θj1 −εn
θj εn
log f
k
rn e dθ ≥
log f k rn eiθ dθ
iθ
En
δ ∞, f k ≥ measEn T rn , f k
4
k
εn δ ∞, f
T rn , f k .
≥
4q
3.22
On the other hand, by the definition of Bα,β r, ∗ and 3.17, we have
θj1 −εn
θj εn
log f k rn eiθ dθ ≤
≤
πrn
Bθj ,θj1 rn , f k
2ωj sin εn ωj
ωj
π 2 Kj,ε 4ωj2 εn
ρε
rn
ωj
rn
,
log rn T rn , f
3.23
where rn ∈
/ E ∪ F, ωj π/θj1 − θj , and Kj,ε is a positive number depending on only j and
ε. Combining 3.22 with 3.23 now yields
εn2 T rn , f k ≤
qπ 2 Kj,ε ρε
ωj
,
r
r
log
r
T
r
,
f
n
n
n
n
ωj2 δ ∞, f k
3.24
so that, together with 3.19 and Lemma 2.9, we have
T
1−2ε
rn , f
k
2
!22ε
qπ 2 2q 1 log rn
Kj,ε ρε
ωj
k
.
r
≤
log
r
log6e
r
log
T
r
,
f
n
n
n
n
ωj2 δ ∞, f k
3.25
Thus μf μf k ≤ max{ρ ε, ωj }/1 − 2ε < ∞ and so Theorem 1.1 follows from
Theorem 1.4.
This completes the proof of Theorem 1.1.
Acknowledgments
The author is extremely grateful to the referees for their many valuable suggestions to
improve the presentation. The project was supported by the National Natural Science
Foundation of China Grant no. 11126351 and the Natural Science Foundation of Fujian
Province, China Grant no. 2010J05003.
16
Abstract and Applied Analysis
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